5,325 research outputs found
Representation of Quantum Mechanical Resonances in the Lax-Phillips Hilbert Space
We discuss the quantum Lax-Phillips theory of scattering and unstable
systems. In this framework, the decay of an unstable system is described by a
semigroup. The spectrum of the generator of the semigroup corresponds to the
singularities of the Lax-Phillips -matrix. In the case of discrete (complex)
spectrum of the generator of the semigroup, associated with resonances, the
decay law is exactly exponential. The states corresponding to these resonances
(eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips
Hilbert space, and therefore all physical properties of the resonant states can
be computed.
We show that the Lax-Phillips -matrix is unitarily related to the
-matrix of standard scattering theory by a unitary transformation
parametrized by the spectral variable of the Lax-Phillips theory.
Analytic continuation in has some of the properties of a method
developed some time ago for application to dilation analytic potentials.
We work out an illustrative example using a Lee-Friedrichs model for the
underlying dynamical system.Comment: Plain TeX, 26 pages. Minor revision
A Chiral Spin Theory in the Framework of an Invariant Evolution Parameter Formalism
We present a formulation for the construction of first order equations which
describe particles with spin, in the context of a manifestly covariant
relativistic theory governed by an invariant evolution parameter; one obtains a
consistent quantized formalism dealing with off-shell particles with spin. Our
basic requirement is that the second order equation in the theory is of the
Schr\"{o}dinger-Stueckelberg type, which exhibits features of both the
Klein-Gordon and Schr\"{o}dinger equations. This requirement restricts the
structure of the first order equation, in particular, to a chiral form. One
thus obtains, in a natural way, a theory of chiral form for massive particles,
which may contain both left and right chiralities, or just one of them. We
observe that by iterating the first order system, we are able to obtain second
order forms containing the transverse and longitudinal momentum relative to a
time-like vector used to maintain covariance of the theory.
This time-like vector coincides with the one used by Horwitz, Piron, and Reuse
to obtain an invariant positive definite space-time scalar product, which
permits the construction of an induced representation for states of a particle
with spin. We discuss the currents and continuity equations, and show that
these equations of motion and their currents are closely related to the spin
and convection parts of the Gordon decomposition of the Dirac current. The
transverse and longitudinal aspects of the particle are complementary, and can
be treated in a unified manner using a tensor product Hilbert space.
Introducing the electromagnetic field we find an equation which gives rise to
the correct gyromagnetic ratio, and is fully Hermitian under the proposed
scalar product. Finally, we show that the original structure of Dirac'sComment: Latex, 61 pages. Minor revisions. To be published in J. Math. Phy
Hypercomplex quantum mechanics
The fundamental axioms of the quantum theory do not explicitly identify the
algebraic structure of the linear space for which orthogonal subspaces
correspond to the propositions (equivalence classes of physical questions). The
projective geometry of the weakly modular orthocomplemented lattice of
propositions may be imbedded in a complex Hilbert space; this is the structure
which has traditionally been used. This paper reviews some work which has been
devoted to generalizing the target space of this imbedding to Hilbert modules
of a more general type. In particular, detailed discussion is given of the
simplest generalization of the complex Hilbert space, that of the quaternion
Hilbert module.Comment: Plain Tex, 11 page
Gravitational Repulsion within a Black-Hole using the Stueckelberg Quantum Formalism
We wish to study an application of Stueckelberg's relativistic quantum theory
in the framework of general relativity. We study the form of the wave equation
of a massive body in the presence of a Schwarzschild gravitational field. We
treat the mathematical behavior of the wavefunction also around and beyond the
horizon (r=2M). Classically, within the horizon, the time component of the
metric becomes spacelike and distance from the origin singularity becomes
timelike, suggesting an inevitable propagation of all matter within the horizon
to a total collapse at r=0. However, the quantum description of the wave
function provides a different understanding of the behavior of matter within
the horizon. We find that a test particle can almost never be found at the
origin and is more probable to be found at the horizon. Matter outside the
horizon has a very small wave length and therefore interference effects can be
found only on a very small atomic scale. However, within the horizon, matter
becomes totally "tachionic" and is potentially "spread" over all space. Small
location uncertainties on the atomic scale become large around the horizon, and
different mass components of the wave function can therefore interfere on a
stellar scale. This interference phenomenon, where the probability of finding
matter decreases as a function of the distance from the horizon, appears as an
effective gravitational repulsion.Comment: 20 pages, 6 figure
Generalized Boltzmann Equation in a Manifestly Covariant Relativistic Statistical Mechanics
We consider the relativistic statistical mechanics of an ensemble of
events with motion in space-time parametrized by an invariant ``historical
time'' We generalize the approach of Yang and Yao, based on the Wigner
distribution functions and the Bogoliubov hypotheses, to find the approximate
dynamical equation for the kinetic state of any nonequilibrium system to the
relativistic case, and obtain a manifestly covariant Boltzmann-type equation
which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU)
equation for indistinguishable particles. This equation is then used to prove
the -theorem for evolution in In the equilibrium limit, the
covariant forms of the standard statistical mechanical distributions are
obtained. We introduce two-body interactions by means of the direct action
potential where is an invariant distance in the Minkowski
space-time. The two-body correlations are taken to have the support in a
relative -invariant subregion of the full spacelike region. The
expressions for the energy density and pressure are obtained and shown to have
the same forms (in terms of an invariant distance parameter) as those of the
nonrelativistic theory and to provide the correct nonrelativistic limit
Estimating proportions of objects from multispectral scanner data
Progress is reported in developing and testing methods of estimating, from multispectral scanner data, proportions of target classes in a scene when there are a significiant number of boundary pixels. Procedures were developed to exploit: (1) prior information concerning the number of object classes normally occurring in a pixel, and (2) spectral information extracted from signals of adjoining pixels. Two algorithms, LIMMIX and nine-point mixtures, are described along with supporting processing techniques. An important by-product of the procedures, in contrast to the previous method, is that they are often appropriate when the number of spectral bands is small. Preliminary tests on LANDSAT data sets, where target classes were (1) lakes and ponds, and (2) agricultural crops were encouraging
A possible mathematics for the unification of quantum mechanics and general relativity
This paper summarizes and generalizes a recently proposed mathematical
framework that unifies the standard formalisms of special relativity and
quantum mechanics. The framework is based on Hilbert spaces H of functions of
four space-time variables x,t, furnished with an additional indefinite inner
product invariant under Poincar\'e transformations, and isomorphisms of these
spaces that preserve the indefinite metric. The indefinite metric is
responsible for breaking the symmetry between space and time variables and for
selecting a family of Hilbert subspaces that are preserved under Galileo
transformations. Within these subspaces the usual quantum mechanics with
Schr\"odinger evolution and t as the evolution parameter is derived.
Simultaneously, the Minkowski space-time is isometrically embedded into H,
Poincar\'e transformations have unique extensions to isomorphisms of H and the
embedding commutes with Poincar\'e transformations. The main new result is a
proof that the framework accommodates arbitrary pseudo-Riemannian space-times
furnished with the action of the diffeomorphism group
Equilibrium Relativistic Mass Distribution for Indistinguishable Events
A manifestly covariant relativistic statistical mechanics of the system of
indistinguishable events with motion in space-time parametrized by an
invariant ``historical time'' is considered. The relativistic mass
distribution for such a system is obtained from the equilibrium solution of the
generalized relativistic Boltzmann equation by integration over angular and
hyperbolic angular variables. All the characteristic averages are calculated.
Expressions for the pressure and the density of events are found and the
relativistic equation of state is obtained. The Galilean limit is considered;
the theory is shown to pass over to the usual nonrelativistic statistical
mechanics of indistinguishable particles.Comment: TAUP-2115-9
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